The operator \((i\nu )\otimes (i\nu )^\flat \colon TM\rightarrow TM\) 
is, at each point \(p\in M\) , the othogonal projection
onto the line spanned by \(i\nu (p)\) . Actually, the covariant
derivative of \(D\nu \)  is related to the Hessian of the function
\(\operatorname{dist}_g(\cdot , \, a)^2\) . There are results in the literature
which provide a much finer description of the Hessian
of \(\operatorname{dist}_g(\cdot , \, a)^2\)  and hence, of \(D\nu \) 
(see e.g. [1]}).
However, the statement given here is enough for our purposes.
We include a proof, for the sake of convenience.
